Properties

Label 756.2.a.b
Level $756$
Weight $2$
Character orbit 756.a
Self dual yes
Analytic conductor $6.037$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.03669039281\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{5} + q^{7} + O(q^{10}) \) \( q - 3 q^{5} + q^{7} + 2 q^{13} - 3 q^{17} - 4 q^{19} - 6 q^{23} + 4 q^{25} - 10 q^{31} - 3 q^{35} - 7 q^{37} - 9 q^{41} + 5 q^{43} - 3 q^{47} + q^{49} - 6 q^{53} + 9 q^{59} + 8 q^{61} - 6 q^{65} + 8 q^{67} + 12 q^{71} - 10 q^{73} + 5 q^{79} - 9 q^{83} + 9 q^{85} - 18 q^{89} + 2 q^{91} + 12 q^{95} + 8 q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −3.00000 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.a.b 1
3.b odd 2 1 756.2.a.e yes 1
4.b odd 2 1 3024.2.a.d 1
7.b odd 2 1 5292.2.a.m 1
9.c even 3 2 2268.2.j.l 2
9.d odd 6 2 2268.2.j.c 2
12.b even 2 1 3024.2.a.ba 1
21.c even 2 1 5292.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.a.b 1 1.a even 1 1 trivial
756.2.a.e yes 1 3.b odd 2 1
2268.2.j.c 2 9.d odd 6 2
2268.2.j.l 2 9.c even 3 2
3024.2.a.d 1 4.b odd 2 1
3024.2.a.ba 1 12.b even 2 1
5292.2.a.a 1 21.c even 2 1
5292.2.a.m 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(756))\):

\( T_{5} + 3 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 3 + T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( -2 + T \)
$17$ \( 3 + T \)
$19$ \( 4 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( 10 + T \)
$37$ \( 7 + T \)
$41$ \( 9 + T \)
$43$ \( -5 + T \)
$47$ \( 3 + T \)
$53$ \( 6 + T \)
$59$ \( -9 + T \)
$61$ \( -8 + T \)
$67$ \( -8 + T \)
$71$ \( -12 + T \)
$73$ \( 10 + T \)
$79$ \( -5 + T \)
$83$ \( 9 + T \)
$89$ \( 18 + T \)
$97$ \( -8 + T \)
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